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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">M</font> being  <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b3">f</font>, <font color="Olive" title="b4">g</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b5">k</font> being  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Olive" title="b2">M</font>,<font color="Olive" title="b5">k</font> &amp; <font color="Olive" title="b4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Olive" title="b2">M</font>,<font color="Olive" title="b5">k</font> &amp; <font color="Olive" title="b3">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Olive" title="b4">g</font>,<font color="Olive" title="b2">M</font> holds <br/> <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Olive" title="b2">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Olive" title="b5">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Olive" title="b2">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Olive" title="b4">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Olive" title="b5">k</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">M</font> being  <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b4">k</font> being  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Olive" title="b1">M</font>,<font color="Olive" title="b4">k</font> &amp; <font color="Olive" title="b3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Olive" title="b1">M</font>,<font color="Olive" title="b4">k</font> &amp; <font color="Olive" title="b2">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Olive" title="b3">g</font>,<font color="Olive" title="b1">M</font> holds <br/> <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Olive" title="b1">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Olive" title="b4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Olive" title="b1">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Olive" title="b3">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Olive" title="b4">k</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b3">k</font> being  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b3">k</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b3">k</font> &amp; <font color="Olive" title="b1">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Olive" title="b2">g</font>,<font color="Maroon" title="c3">M</font> holds <br/> <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Olive" title="b3">k</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">k</font> being  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">k</font> &amp; <font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">k</font> &amp; <font color="Maroon" title="c4">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">M</font> holds <br/> <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Olive" title="b1">k</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">k</font> be   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c4">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">M</font> implies  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> )</span><br/>











<span class="kw">set </span><font color="Maroon" title="c7">t</font> = <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">s</font> = <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font>;<br/>
<span class="kw">assume </span><a NAME="E1:70"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lpspace2.html#K1" title="LPSPACE2:func.1">Lp_Functions</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c4">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c3">M</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span><br/>

<a NAME="E2:70"/>
 ex <font color="Olive" title="b1">f1</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Maroon" title="c4">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">f1</font> &amp;  ex <font color="Olive" title="b2">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">E</font> &amp; <font color="Olive" title="b1">f1</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b2">E</font> &amp; <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Olive" title="b1">f1</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E51">A1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">Df</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E4:70"/><span class="lab"><font color="Green" title="E52">A2</font></span>: 
( <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Df</font> &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c9">Df</font> &amp; <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font> )
 ;<br/>
<a NAME="E5:70"/>
 ex <font color="Olive" title="b1">g1</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Maroon" title="c5">g</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">g1</font> &amp;  ex <font color="Olive" title="b2">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">E</font> &amp; <font color="Olive" title="b1">g1</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b2">E</font> &amp; <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Olive" title="b1">g1</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E51">A1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">Dg</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E7:70"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
( <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Dg</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Dg</font> &amp; <font color="Maroon" title="c5">g</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c10">Dg</font> &amp; <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font> <a href="mesfunc6.html#R3" title="MESFUNC6:pred.3">is_integrable_on</a> <font color="Maroon" title="c3">M</font> )
 ;<br/>
<a NAME="E8:70"/><span class="lab"><font color="Green" title="E54">A4</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D11" title="VALUED_1:def.11">VALUED_1:def 11</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">E1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E10:70"/><span class="lab"><font color="Green" title="E55">A5</font></span>: 
( <font color="Maroon" title="c3">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">E1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">E1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="lpspace1.html#D10" title="LPSPACE1:def.10">LPSPACE1:def 10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">NDf</font> = <font color="Maroon" title="c9">Df</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> , <font color="Maroon" title="c13">NDg</font> = <font color="Maroon" title="c10">Dg</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T66" title="MEASURE1:th.66">MEASURE1:66</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">Ef</font> = <font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">Eg</font> = <font color="Maroon" title="c10">Dg</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">NDf</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">E2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c12">NDf</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c13">NDg</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font>;<br/>
<a NAME="E12:70"/>
( <font color="Maroon" title="c12">NDf</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c13">NDg</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c11">E1</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c3">M</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E7:70"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E10:70"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="ref" href="measure1.html#D13" title="MEASURE1:def.13">MEASURE1:def 13</a></span>;<br/>
<a NAME="E13:70"/><span class="kw">then </span>
( <font color="Maroon" title="c12">NDf</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font> is    <a href="measure1.html#M2" title="MEASURE1:mode.2">measure_zero</a> of <font color="Maroon" title="c3">M</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T70" title="MEASURE1:th.70">MEASURE1:70</a></span>;<br/>
<a NAME="E14:70"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A6</font></span>: 
( <font color="Maroon" title="c3">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">NDf</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#D13" title="MEASURE1:def.13">MEASURE1:def 13</a></span>;<br/>
<a NAME="E15:70"/>
( <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c12">NDf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">Df</font> &amp; <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c13">NDg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c10">Dg</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T48" title="XBOOLE_1:th.48">XBOOLE_1:48</a></span>;<br/>
<a NAME="E16:70"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A7</font></span>: 
( <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c12">NDf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Df</font> &amp; <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c13">NDg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Dg</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<a NAME="E17:70"/>
( <font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <font color="Maroon" title="c13">NDg</font></span>)</span> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <font color="Maroon" title="c11">E1</font> &amp; <font color="Maroon" title="c10">Dg</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">NDf</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Dg</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <font color="Maroon" title="c12">NDf</font></span>)</span> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <font color="Maroon" title="c11">E1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T41" title="XBOOLE_1:th.41">XBOOLE_1:41</a></span>;<br/>
<a NAME="E18:70"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A8</font></span>: 
( <font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">NDf</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c13">NDg</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c11">E1</font> &amp; <font color="Maroon" title="c10">Dg</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">NDf</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">NDf</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c13">NDg</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c11">E1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:70"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="ref" href="xboole_1.html#T41" title="XBOOLE_1:th.41">XBOOLE_1:41</a></span>;<br/>
<a NAME="E19:70"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A9</font></span>: 
( <font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">NDf</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c13">NDg</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span> &amp; <font color="Maroon" title="c10">Dg</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">NDf</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">NDf</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c13">NDg</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T41" title="XBOOLE_1:th.41">XBOOLE_1:41</a></span>;<br/>
<a NAME="E20:70"/>
(  <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c9">Df</font> &amp;  <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c10">Dg</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E7:70"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="ref" href="mesfunc6.html#T48" title="MESFUNC6:th.48">MESFUNC6:48</a></span>;<br/>
<a NAME="E21:70"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">B1</font></span>: 
( <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c9">Df</font> &amp; <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c10">Dg</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E7:70"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E8:70"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="mesfun6c.html#T29" title="MESFUN6C:th.29">MESFUN6C:29</a></span>;<br/>
<a NAME="E22:70"/><span class="lab"><font color="Green" title="E61">B2</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Df</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Dg</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E7:70"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E8:70"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="mesfun6c.html#D6" title="MESFUN6C:def.6">MESFUN6C:def 6</a></span>;<br/>
<a NAME="E23:70"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">B3</font></span>: 
(  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> &amp;  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">Dg</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">NDf</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:70"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="txt" href="#E21:70"><span class="lab"><font color="Green" title="E60">B1</font></span></a>, <a class="ref" href="mesfunc6.html#T89" title="MESFUNC6:th.89">MESFUNC6:89</a></span>;<br/>
<a NAME="E24:70"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/>
<a NAME="E25:70"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">B4</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c9">Df</font></span>)</span> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Dg</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c10">Dg</font></span>)</span> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">NDf</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:70"><span class="lab"><font color="Green" title="E61">B2</font></span></a>, <a class="txt" href="#E18:70"><span class="lab"><font color="Green" title="E58">A8</font></span></a>, <a class="ref" href="xboole_1.html#T49" title="XBOOLE_1:th.49">XBOOLE_1:49</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:70_1"/><span class="lab"><font color="Green" title="E64">B5</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font></span><br/><a NAME="E2:70_1"/><span class="lab"><font color="Green" title="E65">B6</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></span>;<br/><a NAME="E3:70_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">NDf</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c13">NDg</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">E1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T34" title="XBOOLE_1:th.34">XBOOLE_1:34</a></span>;<br/><a NAME="E4:70_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">B7</font></span>: 
( <font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">E1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font> &amp; <font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">E1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1"><span class="lab"><font color="Green" title="E64">B5</font></span></a>, <a class="txt" href="#E25:70"><span class="lab"><font color="Green" title="E63">B4</font></span></a>, <a class="txt" href="#E19:70"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/><a NAME="E5:70_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1"><span class="lab"><font color="Green" title="E64">B5</font></span></a>, <a class="txt" href="#E25:70"><span class="lab"><font color="Green" title="E63">B4</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/><a NAME="E6:70_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c6">k</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c6">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:70"><span class="lab"><font color="Green" title="E58">A8</font></span></a>, <a class="txt" href="#E25:70"><span class="lab"><font color="Green" title="E63">B4</font></span></a>, <a class="txt" href="#E1:70_1"><span class="lab"><font color="Green" title="E64">B5</font></span></a>, <a class="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E65">B6</font></span></a>, <a class="ref" href="mesfun6c.html#D6" title="MESFUN6C:def.6">MESFUN6C:def 6</a></span>;<br/><a NAME="E7:70_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c6">k</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font></span>)</span></span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c6">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T18" title="VALUED_1:th.18">VALUED_1:18</a></span>;<br/><span class="kw">hence </span><a NAME="E8:70_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">Df</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">NDg</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c11">E1</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:70"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="txt" href="#E4:70_1"><span class="lab"><font color="Green" title="E66">B7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E27:70"/>
 <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="mesfun6c.html#K4" title="MESFUN6C:func.4">to_power</a> <font color="Maroon" title="c6">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:70"><span class="lab"><font color="Green" title="E62">B3</font></span></a>, <a class="txt" href="#E25:70"><span class="lab"><font color="Green" title="E63">B4</font></span></a>, <a class="txt" href="#E18:70"><span class="lab"><font color="Green" title="E58">A8</font></span></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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